A nonlinear lower bound on the practical combinational complexity

AbstractAn infinite sequence F = {fn}n = 1∞ of one-output Boolean functions with the following two properties is constructed: 1.(1)fn can be computed by a Boolean circuit with O(n) gates.2.(2)For any positive, nondecreasing, and unbounded function h : N → R, each Boolean circuit having an mh(m) separator requires a nonlinear number Ω(nh(n)) of gates to compute fn (e.g., each planar Boolean circuit requires Ω(n2) gates to compute fn).Thus, one can say that fn has linear combinational complexity and a nonlinear practical combinational complexity because the constant-degree parallel architectures used in practice have separators in O(mlog2 m)

Recent Advances in Graph Partitioning

We survey recent trends in practical algorithms for balanced graph partitioning together with applications and future research directions

Energy-Efficient Algorithms on Mesh-Connected Systems with Additional Communication Links.

Energy consumption has become a critical factor constraining the design of massively parallel computers, necessitating the development of new models and energy-efficient algorithms. In this work we take a fundamental abstract model of massive parallelism, the mesh-connected computer, and extend it with additional communication links motivated by recent advances in on-chip photonic interconnects. This new means of communication with optical signals rather than electrical signals can reduce the energy and/or time of calculations by providing faster communication between distant processing elements. Processors are arranged in a two-dimensional grid with wire connections between adjacent neighbors and an additional one or two layers of noncrossing optical connections. Varying constraints on the layout of optics affect how powerful the model can be. In this dissertation, three optical interconnection layouts are defined: the optical mesh, the optical mesh of trees, and the optical pyramid. For each layout, algorithms for solving important problems are presented. Since energy usage is an important factor, running times are given in terms of a peak-power constraint, where peak power is the maximum number of processors active at any one time. These results demonstrate advantages of optics in terms of improved time and energy usage over the standard mesh computer without optics. One of the most significant results shows an optimal nonlinear time/peak-power tradeoff for sorting on the optical pyramid. This work shows asymptotic theoretical limits of computation and energy usage on an abstract model which takes physical constraints and developing interconnection technology into account.PhDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/102474/1/ppoon_1.pd

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum